Order Statistics Comp 122, Spring 2004

Order Statistic ith order statistic: ith smallest element of a set of n elements. Minimum: first order statistic. Maximum: nth order statistic. Median: “half-way point” of the set. Unique, when n is odd – occurs at i = (n+1)/2. Two medians when n is even. Lower median, at i = n/2. Upper median, at i = n/2+1. For consistency, “median” will refer to the lower median. Comp 122

Selection Problem Selection problem: Input: A set A of n distinct numbers and a number i, with 1 i  n. Output: the element x  A that is larger than exactly i – 1 other elements of A. Can be solved in O(n lg n) time. How? We will study faster linear-time algorithms. For the special cases when i = 1 and i = n. For the general problem. Comp 122

Minimum (Maximum) No. of comparisons: n – 1. Minimum (A) 1. min  A[1] 2. for i  2 to length[A] 3. do if min > A[i] 4. then min  A[i] 5. return min Maximum can be determined similarly. T(n) = (n). No. of comparisons: n – 1. Can we do better? Why not? Minimum(A) has worst-case optimal # of comparisons. Comp 122

Problem Minimum (A) 1. min  A[1] 2. for i  2 to length[A] 3. do if min > A[i] 4. then min  A[i] 5. return min Average for random input: How many times do we expect line 4 to be executed? X = RV for # of executions of line 4. Xi = Indicator RV for the event that line 4 is executed on the ith iteration. X = i=2..n Xi E[Xi] = 1/i. How? Hence, E[X] = ln(n) – 1 = (lg n). Comp 122

Simultaneous Minimum and Maximum Some applications need to determine both the maximum and minimum of a set of elements. Example: Graphics program trying to fit a set of points onto a rectangular display. Independent determination of maximum and minimum requires 2n – 2 comparisons. Can we reduce this number? Yes. Comp 122

Simultaneous Minimum and Maximum Maintain minimum and maximum elements seen so far. Process elements in pairs. Compare the smaller to the current minimum and the larger to the current maximum. Update current minimum and maximum based on the outcomes. No. of comparisons per pair = 3. How? No. of pairs  n/2. For odd n: initialize min and max to A[1]. Pair the remaining elements. So, no. of pairs = n/2. For even n: initialize min to the smaller of the first pair and max to the larger. So, remaining no. of pairs = (n – 2)/2 < n/2. Comp 122

Simultaneous Minimum and Maximum Total no. of comparisons, C  3n/2. For odd n: C = 3n/2. For even n: C = 3(n – 2)/2 + 1 (For the initial comparison). = 3n/2 – 2 < 3n/2. Comp 122

General Selection Problem Seems more difficult than Minimum or Maximum. Yet, has solutions with same asymptotic complexity as Minimum and Maximum. We will study 2 algorithms for the general problem. One with expected linear-time complexity. A second, whose worst-case complexity is linear. Comp 122

Selection in Expected Linear Time Modeled after randomized quicksort. Exploits the abilities of Randomized-Partition (RP). RP returns the index k in the sorted order of a randomly chosen element (pivot). If the order statistic we are interested in, i, equals k, then we are done. Else, reduce the problem size using its other ability. RP rearranges the other elements around the random pivot. If i < k, selection can be narrowed down to A[1..k – 1]. Else, select the (i – k)th element from A[k+1..n]. (Assuming RP operates on A[1..n]. For A[p..r], change k appropriately.) Comp 122

Randomized Quicksort: review Quicksort(A, p, r) if p < r then q := Rnd-Partition(A, p, r); Quicksort(A, p, q – 1); Quicksort(A, q + 1, r) fi Rnd-Partition(A, p, r) i := Random(p, r); A[r]  A[i]; x, i := A[r], p – 1; for j := p to r – 1 do if A[j]  x then i := i + 1; A[i]  A[j] fi od; A[i + 1]  A[r]; return i + 1 A[p..r] 5 A[p..q – 1] A[q+1..r] Partition 5  5  5 Comp 122

Randomized-Select Randomized-Select(A, p, r, i) // select ith order statistic. 1. if p = r 2. then return A[p] 3. q  Randomized-Partition(A, p, r) 4. k  q – p + 1 5. if i = k 6. then return A[q] 7. elseif i < k 8. then return Randomized-Select(A, p, q – 1, i) 9. else return Randomized-Select(A, q+1, r, i – k) Comp 122

Analysis Worst-case Complexity: Average-case Complexity: (n2) – As we could get unlucky and always recurse on a subarray that is only one element smaller than the previous subarray. Average-case Complexity: (n) – Intuition: Because the pivot is chosen at random, we expect that we get rid of half of the list each time we choose a random pivot q. Why (n) and not (n lg n)? Comp 122

Average-case Analysis Define Indicator RV’s Xk, for 1  k  n. Xk = I{subarray A[p…q] has exactly k elements}. Pr{subarray A[p…q] has exactly k elements} = 1/n for all k = 1..n. Hence, E[Xk] = 1/n. Let T(n) be the RV for the time required by Randomized-Select (RS) on A[p…q] of n elements. Determine an upper bound on E[T(n)]. (9.1) Comp 122

Average-case Analysis A call to RS may Terminate immediately with the correct answer, Recurse on A[p..q – 1], or Recurse on A[q+1..r]. To obtain an upper bound, assume that the ith smallest element that we want is always in the larger subarray. RP takes O(n) time on a problem of size n. Hence, recurrence for T(n) is: For a given call of RS, Xk =1 for exactly one value of k, and Xk = 0 for all other k. Comp 122

Average-case Analysis (by linearity of expectation) (by Eq. (C.23)) (by Eq. (9.1)) Comp 122

Average-case Analysis (Contd.) The summation is expanded If n is odd, T(n – 1) thru T(n/2) occur twice and T(n/2) occurs once. If n is even, T(n – 1) thru T(n/2) occur twice. Comp 122

Average-case Analysis (Contd.) We solve the recurrence by substitution. Guess E[T(n)] = O(n). Thus, if we assume T(n) = O(1) for n < 2c/(c – 4a), we have E[T(n)] = O(n). Comp 122

Selection in Worst-Case Linear Time Algorithm Select: Like RandomizedSelect, finds the desired element by recursively partitioning the input array. Unlike RandomizedSelect, is deterministic. Uses a variant of the deterministic Partition routine. Partition is told which element to use as the pivot. Achieves linear-time complexity in the worst case by Guaranteeing that the split is always “good” at each Partition. How can a good split be guaranteed? Comp 122

Guaranteeing a Good Split We will have a good split if we can ensure that the pivot is the median element or an element close to the median. Hence, determining a reasonable pivot is the first step. Comp 122

Choosing a Pivot Median-of-Medians: Divide the n elements into n/5 groups.  n/5 groups contain 5 elements each. 1 group contains n mod 5 < 5 elements. Determine the median of each of the groups. Sort each group using Insertion Sort. Pick the median from the sorted list of group elements. Recursively find the median x of the n/5 medians. Recurrence for running time (of median-of-medians): T(n) = O(n) + T(n/5) + …. Comp 122

Algorithm Select Determine the median-of-medians x (using the procedure on the previous slide.) Partition the input array around x using the variant of Partition. Let k be the index of x that Partition returns. If k = i, then return x. Else if i < k, then apply Select recursively to A[1..k–1] to find the ith smallest element. Else if i > k, then apply Select recursively to A[k+1..n] to find the (i – k)th smallest element. (Assumption: Select operates on A[1..n]. For subarrays A[p..r], suitably change k. ) Comp 122

Worst-case Split Arrows point from larger to smaller elements. n/5 groups of 5 elements each. Elements < x n/5th group of n mod 5 elements. Median-of-medians, x Elements > x Comp 122

Worst-case Split Assumption: Elements are distinct. Why? At least half of the n/5 medians are greater than x. Thus, at least half of the n/5 groups contribute 3 elements that are greater than x. The last group and the group containing x may contribute fewer than 3 elements. Exclude these groups. Hence, the no. of elements > x is at least Analogously, the no. of elements < x is at least 3n/10–6. Thus, in the worst case, Select is called recursively on at most 7n/10+6 elements. Comp 122

Recurrence for worst-case running time T(Select)  T(Median-of-medians) +T(Partition) +T(recursive call to select) T(n)  O(n) + T(n/5) + O(n) + T(7n/10+6) = T(n/5) + T(7n/10+6) + O(n) Assume T(n)  (1), for n  140. T(Median-of-medians) T(Partition) T(recursive call) Comp 122

Solving the recurrence To show: T(n) = O(n)  cn for suitable c and all n > 0. Assume: T(n)  cn for suitable c and all n  140. Substituting the inductive hypothesis into the recurrence, T(n)  c n/5 + c(7n/10+6)+an  cn/5 + c + 7cn/10 + 6c + an = 9cn/10 + 7c + an = cn +(–cn/10 + 7c + an)  cn, if –cn/10 + 7c + an  0. n/(n–70) is a decreasing function of n. Verify. Hence, c can be chosen for any n = n0 > 70, provided it can be assumed that T(n) = O(1) for n  n0. Thus, Select has linear-time complexity in the worst case. –cn/10 + 7c + an  0  c  10a(n/(n – 70)), when n > 70. For n  140, c  20a. Comp 122